Frequency concentration and location lengths for the Anderson model at small disorders (Q1411685)

The paper deals with the random operator \(H = \Delta_{Z^d} + \lambda \omega\), \(\lambda >0\), \(d = 1,2,\) acting on the space \(\ell^2 (Z^d)\). Here, for \(u \in \ell^2 (Z^d)\), \(\left(\Delta_{Z^d} u \right) (n) = \sum_{m: \;|m - n|=1} u(m),\) i.e., \(\Delta_{Z^d}\) is a version of the discrete lattice Laplacian, \(\omega: \ell^2 (Z^d) \rightarrow \ell^2 (Z^d)\) is the multiplication operator \((\omega u)(n) = \omega_n u(n)\), and \(\{\omega_n\}_{n \in Z^d}\) is a sequence of independent identically distributed bounded random variables with zero mean and unit variance. The operator \(H\) is considered as a discretization of the Anderson Hamiltonian \(H_A = - \Delta + \lambda V(x),\) acting in \(L^2 ({\mathbb R}^d)\), where \(\Delta\) is the Laplacian and \(V\) is a real Gaussian process. The main results of the paper are contained in two theorems describing eigenfunctions of \(H\) and their Fourier transforms in the limit \(\lambda \rightarrow 0\) for \(d=1\) and \(2\).